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3 4. Evaluate. fovz+ +e" + coS %) dx I...

Question

3 4. Evaluate. fovz+ +e" + coS %) dx I

3 4. Evaluate. fovz+ +e" + coS %) dx I



Answers

Evaluate the integrals.
$\int \cos 3 x \cos 4 x d x$

Section eight Got to We're looking at Problem number 57 the indefinite integral co sign of three x times the co sign of four X. So in order to work this we're going to use the identity that were given is that the co sign of M X time to co sign and in X is going to be equal to 1/2. I'm the co sign of M minus in of X plus the co sign of M plus end of X. So this is the identity that's gonna allow us to integrate this because it takes this product and it breaks up into two separate intervals. So in this case, I'm looking at this solution with him equal to three and in equal to four. So this integral converts to 1/2 the integral of so that's going to be the co sign, uh, minus X plus the co sign of seven x DX. Now you know that the Cho san isn't even function, So the co sign of minus X is equal to the coastline of X. So with his converts to this is 1/2 coastline of X plus a coastline of seven x dx you integrate the co sign and you get the sine function. So this is going to be 1/2 sign of X plus the sign of seven X times 1/7 plus a constant integration. And sorry about that. There is a 1/2 that is involved. So let me just back up when I got to this spot right here, just to be clear. Minor with magic mistake. You've got the 1/2 is in front here. So with this 1/2 right here, this becomes plus 1/2 the sign of seven x times 1/7 plus a constant of integration. So this final answer is 1/2 sign of X plus one 14th sign of seven x plus a constant of integrate.

We want to integrate free concern for X. Ah, couple minutes for these questions You were used. Constitution meant that let u equals 24 x When we differentiate you, we differentiate for X. We get full and nd x so we need to get a four here. So what we do is I bring the tree in front. I divide by four so I can't introduce it for here. He x and I have course mindful Xia. Now I can substitute This will be you and this will be Do you So I have assigned you the U where we integrate co sign will get signed Let's see And we cannot leave our answer in you We have to replace back with ex So I have tree Quander sign four x plus c were sees a constant

Here. And so when you take the entire day of a trick function, we have to think that too. What exhibits aren't so the jury to live side disco aside and the coast is naked. So well, you think about the idea of the opposite operation of a dream. It you know the answer. You're of sign of axes in a negative. So negative 1/3. What? And similarly, the unpolluted of coastline Bax is going to sign bucks so minus one for side on since it is a indefinite rolled me, remember to add right plus C and this is out into

Since we're given the integral of negative three co sign of X plus four seeking squared of X. I just try to think of because I don't want to memorize everything. I try to think of some function whose derivative will give me co sign. And I know that the derivative of Sine gives me co sign. So the constant of -3 just goes along for the ride. So it's true that the derivative of -3 sine of X is equal to negative three co sign effects. And I do the same thought process on what story who give me seeking squared and I have that memorized to be tangent of X. Um And again the coefficient just goes along for the ride and I know I'm right because the derivative of four tangent of X is equal to four seconds squared. Um But you also need your plus C. Because the derivative of a constant is zero. And we don't write plus zero up here. And a quick internet search should help you make that connection between derivatives and intervals. So this is the correct answer.


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